Fast computing of conformal mapping and its inverse of bounded multiply connected regions onto second, third and fourth categories of koebe’s canonical slit regions

Abstract

This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded multiply connected region onto a disk with spiral slits region Ω1. This extends the methods that have recently been given for mappings onto annulus with spiral slits region Ω2, spiral slits region Ω3, and straight slits region Ω4 but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto all four regions Ω1, Ω2, Ω3, and Ω4 as well as their inverses. The integral equations are solved numerically using combination of Nyström method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is O((m+ 1) n) , where m+ 1 is the multiplicity of the multiply connected region and n is the number of nodes on each boundary component. Previous algorithms require O((m+ 1) 3n3) operations. The algorithm is tested on several test regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method.